I like to think of brick stitch as being the opposite of peyote stitch. The stitches are each done very differently, but unless you look at the thread paths (usually under magnification), you cannot tell the difference.
But of course there are differences: As you're stitching peyote stitch, the sides of your work (where the ends of the rows are) are smooth and even, and the beads of each row form a zig-zag shape (i.e., up beads and down beads). As you're stitching brick stitch, the sides of the work (where the ends of the rows are) are uneven, forming a zig-zag shape, while the rows are flat and even.
A second difference is the way the beads lie. In peyote stitch the hole in the beads face the beginning and end of the row. In brick stitch, the beads sit so the holes are facing up and down.
The third difference is of course the thread path, i.e., it's a different stitch from peyote! So let's give it a try.
There are three ways to begin brick stitch. One way is to build a base row using ladder stitch. This is fine, but it's a different stitch, and therefore the tension is not the same as it will be in brick stitch.
Another way is to build it off the flat side of a piece of peyote work. See this project for an example.
The way I'm going to show here is a traditional method used by the African originators of the stitch. I read about it in a book some years ago, and I think the book was by either Diane Fitzgerald or Virginia Blakelock. The author was studying old pieces of beadwork and discovered the ancient stitchers did not use ladder stitch to begin brick stitch.
Please note that if you're following a pattern for your project, you have to follow it as you're stithing the method below.
Anyhoo, here it is.
1. Put a needle on a conditioned length of thread. You can use a stop bead if you'd like. String three beads. In the photos below, I'm using double Delicas. Go back down through the first bead strung so that the tail and the working thread are exiting from the same bead in the same direction.
2. We'll be building rows 1 AND 2 of your brick stitch project (cool, huh?)--don't forget to follow your pattern if you're using one. If you're right-handed, your working thread should be on the right and you should be holding the tail with your left hand. If you're left-handed, it's the opposite, of course.
3. Pick up a new bead, and right-handers will go up through the bead to the top right, and left-handers will go up through the bead on the top left. I'm right-handed, so the photos reflect this.
4. Continue to work to the right or left based on your dominant hand.
5. Now pick up another bead and go down through the bead on the bottom right or bottom left.
6. Continue steps 3-5 until rows 1 and 2 contain as many beads as desired. If your row 1 has fewer beads in it (as in the photo below), the working thread will be exiting the last bead added to row 1 (see bottom right bead in photo below). If my hypothetical pattern called for rows 1 and 2 to each have four beads, the thread would be exiting the top right bead.
7. My project here has three beads in row 1, four beads in row 2, and five beads in row 3. Rows 1 and 2 are complete (see photo above), but my thread is in the wrong place since it's exiting a row 1 bead. I have to reposition my thread so that it exits a bead at either end of row 2. See the photo below. Of course, to reposition, you travel through the beads, not across the outsides of them. That would be gross.
8. Now we're ready to begin brick stitch proper. But let's notice a few things before continuing. The rows in the photograph above are horizontal, so notice how the side of the piece are zig-zagging, and notice how the top of the row is flat and smooth. Compare this with a piece of peyote being stitched in the identical direction. Next, notice the holes in the beads in the photos above are facing the top (and bottom) of the project, whereas in peyote, they holes are facing the ends of the rows (or edges of the project). Last, notice the thread connecting the tops of the beads. I call those connections "bridges." We'll use those bridges to work brick stitch.
Brick stitch is picking up one or two beads, taking the needle under a bridge, and then taking the needle back up through the last (or only) bead added. At the beginning of EVERY row of brick stitch (no matter if it's an increase row, a decrease row, or an even row), you pick up two beads.
9. My project has five beads in row 3, so that's an increase from row 2. So, I'm picking up two beads, and because this is an increase row, I'm taking thread under the FIRST bridge.
Now I'm taking the thread up through the last bead added, which in the photos above and below is the bead on the left in row 3.
10. Now I'm ready to make another stitch. Pick up one bead and take the thread under the next bridge. Then take the needle up through the bead just added.
11. Again, my row 3 is an increase row. This means I started the row by going under the first bridge, and it means I have to use the last bridge twice.
12. To work a row even (i.e., the row about to be worked has the same number of beads as the last row), start the row with two beads (as with the beginning of every row) and go under the second bridge (as we did for the increase row), but only go under the last bridge ONCE.
13. To work a decrease row, begin the row with two beads (as with the beginning of every row), but this time go under the SECOND bridge. At the end of this decrease row, go under the last bridge ONCE.
In brick stitch, every stitch takes two needle passes: the first pass is under the bridge, and the second pass is through the last (or only) bead added. Your thread has to be exiting a bead before you're ready to make the next stitch.